Maximal-time criteria for a generalized derivative nonlinear Schrödinger equation

Senyue Luo; Meilan Qiu; Zhiming Tan; Senyue Luo; Meilan Qiu; Zhiming Tan

doi:10.3934/math.20261098

AIMS Mathematics

2026, Volume 11, Issue 8: 27448-27467. doi: 10.3934/math.20261098

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Maximal-time criteria for a generalized derivative nonlinear Schrödinger equation

1.
Department of Basic Education, The Open University of Guangdong, Guangdong Polytechnic Institute, Guangzhou, Guangdong, 510091, China
2.
School of Mathematics and Statistics, Huizhou University, Huizhou, Guangdong, 516007, China

Received: 24 May 2026 Revised: 19 July 2026 Accepted: 31 July 2026 Published: 31 August 2026
MSC : 35B30, 35C07, 35Q55

We study a generalized derivative nonlinear Schrödinger equation on a bounded interval under homogeneous Dirichlet boundary conditions. Starting from the local $ H^2 $ well-posedness theory, we derive a quantitative $ H^2 $ energy inequality in physical space. The estimate yields a genuine continuation criterion at a finite endpoint, divergence of the controlling $ W^{1, \infty} $ time integral whenever the maximal time is finite, the $ H^2 $ blow-up alternative, a quantitative lower blow-up rate, and an explicit lower bound for the lifespan in terms of the initial $ H^2 $ norm. The key mechanism is a derivative-loss cancellation in the highest-order transport term. Thus, the local existence theory is supplemented by explicit maximal-time information and computable growth scales.
- generalized derivative nonlinear Schrödinger equation,
- maximal-time criteria,
- blow-up alternative,
- $ H^2 $ energy estimates
Citation: Senyue Luo, Meilan Qiu, Zhiming Tan. Maximal-time criteria for a generalized derivative nonlinear Schrödinger equation[J]. AIMS Mathematics, 2026, 11(8): 27448-27467. doi: 10.3934/math.20261098

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Abstract

We study a generalized derivative nonlinear Schrödinger equation on a bounded interval under homogeneous Dirichlet boundary conditions. Starting from the local $ H^2 $ well-posedness theory, we derive a quantitative $ H^2 $ energy inequality in physical space. The estimate yields a genuine continuation criterion at a finite endpoint, divergence of the controlling $ W^{1, \infty} $ time integral whenever the maximal time is finite, the $ H^2 $ blow-up alternative, a quantitative lower blow-up rate, and an explicit lower bound for the lifespan in terms of the initial $ H^2 $ norm. The key mechanism is a derivative-loss cancellation in the highest-order transport term. Thus, the local existence theory is supplemented by explicit maximal-time information and computable growth scales.

References

[1]	M. Hayashi, T. Ozawa, Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differ. Equ., 261 (2016), 5424–5445. https://doi.org/10.1016/j.jde.2016.08.018 doi: 10.1016/j.jde.2016.08.018
[2]	D. J. Kaup, A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 19 (1978), 798–801. https://doi.org/10.1063/1.523737 doi: 10.1063/1.523737
[3]	K. Mio, T. Ogino, K. Minami, S. Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Jpn., 41 (1976), 265–271. https://doi.org/10.1143/JPSJ.41.265 doi: 10.1143/JPSJ.41.265
[4]	E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys., 16 (1976), 321–334. https://doi.org/10.1017/S0022377800020249 doi: 10.1017/S0022377800020249
[5]	J. Moses, B. A. Malomed, F. W. Wise, Self-steepening of ultrashort optical pulses without self-phase-modulation, Phys. Rev. A, 76 (2007), 021802(R). https://doi.org/10.1103/PhysRevA.76.021802 doi: 10.1103/PhysRevA.76.021802
[6]	M. Tsutsumi, I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259–277. https://doi.org/10.24546/0100499343 doi: 10.24546/0100499343
[7]	N. Hayashi, T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14–36. https://doi.org/10.1016/0167-2789(92)90185-P doi: 10.1016/0167-2789(92)90185-P
[8]	N. Hayashi, T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488–1503. https://doi.org/10.1137/S0036141093246129 doi: 10.1137/S0036141093246129
[9]	N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823–833. https://doi.org/10.1016/0362-546X(93)90071-Y doi: 10.1016/0362-546X(93)90071-Y
[10]	T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137–164. https://doi.org/10.1512/iumj.1996.45.1962 doi: 10.1512/iumj.1996.45.1962
[11]	H. A. Biagioni, F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649–3659. https://doi.org/10.1090/S0002-9947-01-02754-4 doi: 10.1090/S0002-9947-01-02754-4
[12]	C. X. Miao, Y. F. Wu, G. X. Xu, Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$, J. Differ. Equ., 251 (2011), 2164–2195. https://doi.org/10.1016/j.jde.2011.07.004 doi: 10.1016/j.jde.2011.07.004
[13]	H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561–580. https://doi.org/10.57262/ade/1366031032 doi: 10.57262/ade/1366031032
[14]	A. Grünrock, S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890–1920. https://doi.org/10.1137/070689139 doi: 10.1137/070689139
[15]	S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., 2006 (2006), 096763. https://doi.org/10.1155/IMRN/2006/96763 doi: 10.1155/IMRN/2006/96763
[16]	Z. H. Guo, Y. F. Wu, Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{1/2}(\mathbb{R})$, Discrete Contin. Dyn. Syst., 37 (2017), 257–264. https://doi.org/10.3934/dcds.2017010 doi: 10.3934/dcds.2017010
[17]	Y. F. Wu, Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space, Anal. PDE, 6 (2013), 1989–2002. https://doi.org/10.2140/apde.2013.6.1989 doi: 10.2140/apde.2013.6.1989
[18]	N. Hayashi, P. I. Naumkin, Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited, Discrete Contin. Dyn. Syst., 3 (1997), 383–400. https://doi.org/10.3934/dcds.1997.3.383 doi: 10.3934/dcds.1997.3.383
[19]	D. M. Ambrose, G. Simpson, Local existence theory for derivative nonlinear Schrödinger equations with noninteger power nonlinearities, SIAM J. Math. Anal., 47 (2015), 2241–2264. https://doi.org/10.1137/140955227 doi: 10.1137/140955227
[20]	F. Linares, G. Ponce, G. N. Santos, On a class of solutions to the generalized derivative Schrödinger equations, Acta Math. Sin. Engl. Ser., 35 (2019), 1057–1073. https://doi.org/10.1007/s10114-019-7540-4 doi: 10.1007/s10114-019-7540-4
[21]	K. Fujiwara, T. Ozawa, On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation, Evol. Equ. Control Theory, 7 (2018), 275–280. https://doi.org/10.3934/eect.2018013 doi: 10.3934/eect.2018013
[22]	T. Meškauskas, F. Ivanauskas, Initial boundary-value problems for derivative nonlinear Schroedinger equation. Justification of two-step algorithm, Nonlinear Anal. Model. Control, 7 (2002), 69–104. https://doi.org/10.15388/NA.2002.7.2.15195 doi: 10.15388/NA.2002.7.2.15195
[23]	B. Pineau, M. A. Taylor, Global well-posedness for the generalized derivative nonlinear Schrödinger equation, Nonlinearity, 38 (2025), 055022. https://doi.org/10.1088/1361-6544/add076 doi: 10.1088/1361-6544/add076
[24]	M. Shan, Global asymptotic behavior of solutions to the generalized derivative nonlinear Schrödinger equation, Math. Z., 311 (2025), 76. https://doi.org/10.1007/s00209-025-03880-x doi: 10.1007/s00209-025-03880-x

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